The present invention relates generally to microelectromechanical systems (MEMS). MEMS are devices formed from miniaturized components operatively arranged on a substrate. These components are constructed through the use of lithographic and other micro-fabrication technologies to yield, for example, sensors and actuators.
Many common micromechanical structures are based on the reaction (e.g., oscillation, deflection or torsion) of a beam structure to an applied force. Such beam structures usually have, or are modeled to have, a rectangular cross section. However, the degree to which a beam is actually “rectangular” depends on the anisotropy of the etching method used to form it. Beams are used in the suspension of rigid plates, as lateral oscillators, or as cantilever devices. They are a natural choice for bearing-less motion detectors. Of particular note, MEMS increasingly use beams within resonator structures as part of clock and signal filtering circuits.
Single crystal semiconductors, such as silicon, are the obvious material of choice for the fabrication of resonator beams. Such materials have excellent mechanical strength and high intrinsic quality factor. Furthermore, the formation and processing of silicon-based materials are well-developed fields of endeavor drawing upon decades of experience from the integrated circuit industry.
Using polycrystalline silicon (“Poly Si”), for example, one may design resonators having great flexibility in geometry. However, the simple, but commonly used, bending beam and lateral oscillating beam structures will serve to illustrate not only some of the performance concerns associated with conventional resonators, but also the precepts of the present invention that follow.
Looking at FIG. 1, a bending beam structure is formed by suspending a length of beam 1 having a rectangular cross section above a semiconductor substrate 3 by means of end anchors 5. Typically, an actuating electrode (not shown) is associated with the beam, i.e., placed in electrostatic field proximity to the beam. The beam is excited by an electrostatic field induced by the electrode and mechanically vibrates in sympathy with oscillations in the electrostatic field.
When a force is applied to the surface of a beam, that surface is said to be stressed. The average value of this stress, σ, may be expressed as the loading force, F, divided by the area, A, over which it is applied, or:
  σ  =      F    A  
When subjected to a stress, materials literally get pushed (or pulled) out of shape. Strain, ε, is a measure of this deformation, within the elastic limits of the material, and equals the change in length, ΔL, divided by the original length, LO, or:
  ɛ  =            Δ      ⁢                          ⁢      L              L      O      
Most materials of interest deform linearly with load. Since load is proportional to stress and deformation is proportional to strain, stress and strain are linearly related. The proportionality constant that relates these two measures is known as the elastic modulus or Young's modulus for the material and is given the symbol “E.” Young's moduli are known for a great range of materials.
The mechanical stiffness, kM, of a beam, as calculated with respect to the oscillation direction parallel to the width of the beam “w,” is proportional to its Young's modulus, E, and certain measures of its geometry, including for a beam with a rectangular cross section; length, “L,” and height, “h.”
                              k          M                ≈                              E            ·            h            ·                          w              3                                            L            3                                              EQUATION        ⁢                                  ⁢        1            
As is well understood, the Young's modulus for most materials of interest changes with temperature according to known thermal coefficients (αE). For example, Poly Si has a thermal coefficient of 30 ppm/K°. Furthermore, the geometry of a beam structure also changes with temperature, generally expanding with increasing in temperature. Again, as an example, Poly Si has a thermal expansion coefficient, αexp, of 2.5 ppm/K°.
For some beam designs and related modeling purposes, and given a material with an isotropic thermal coefficient, the effect of thermal expansion on the width of the beam is essentially offset by the effect of thermal expansion on the length of the beam, thus resulting in a remaining linear effect on the height of the beam.
Setting aside electrostatic forces, the resonance frequency (f) of a beam may thus be defined under these assumptions by the equation:
                    f        ≈                              1                          2              ·              π                                ·                                                    k                M                                            m                eff                                                                        EQUATION        ⁢                                  ⁢        2            
where meff is the effective mass of the beam, constant over temperature.
Given the critical nature of a beam's resonance frequency to the overall performance of the resonator, it must remain relatively stable over a range of operating temperatures. In view of the relationship set forth in EQUATION 2, frequency will remain constant only if the mechanical stiffness remains constant. This, however, will not normally be the case as thermally induced changes to the Young's modulus tend to change in the mechanical stiffness of the beam. Accordingly, some external influence is required to “compensate” for the inevitable changes in resonance frequency due to variations in temperature.
Prior attempts have been made to address the issue of resonant beam frequency stabilization in the presence of changing temperature. See, for example, Wan-Thai Hsu, Stiffness-Compensated Temperature Insensitive Micromechanical Resonators, MEMS 2002 (-7803-7185-February 2002 IEEE). Such attempts have, however, focused on the issue of vertical oscillation compensation and have prescribing the remedial use of gold or similar materials that are incompatible with CMOS integration.
For other beam designs and related modeling purposes, the frequency (f) of a resonance beam having a rectangular cross section may be expressed by the following equation:
                    f        ≈                              t                          L              2                                ⁢                                    E              ρ                                ⁢                      (                          1              +                                                                    L                    2                                                        7                    ⁢                                          t                      2                                                                      ⁢                S                                      )                                              EQUATION        ⁢                                  ⁢        3            where “ρ” is the density of the material forming the beam, and “S” is an elastic strain applied to the beam.
As temperature rises, both L and t increase due to thermal expansion, but the effect of the changes in L dominate due to the fact that L is much, much greater than t. As a result, the frequency tends to decrease as temperature increases, and vice versa. Also apparent from the foregoing equation, compressive strain applied to the beam with increasing temperature will enhance frequency sensitivity as a function of temperature. Conversely, tensile strain applied to the beam with increasing temperature will retard frequency sensitivity as a function of temperature. Such conditions can be better understood by first assuming a desired relationship wherein the change in frequency, d(f) as a function of the change in temperature, d(T) is equal to 0. Substituting and equating expressions yields:
                                          α            exp                    ⁡                      (                          1              +                                                                    L                    2                                                        7                    ⁢                                          t                      2                                                                      ⁢                S                                      )                          =                                            L              2                                      7              ⁢                              t                2                                              ⁢                                    ⅆ              S                                      ⅆ              T                                                          EQUATION        ⁢                                  ⁢        4            
For most practical situations, the applied strain, S, will be much, much less than one. Under such assumptions, the relationship described in EQUATION 4 becomes:
                                          ⅆ            S                                ⅆ            T                          =                                            7              ⁢                              t                2                                                    L              2                                ⁢                      α            exp                                              EQUATION        ⁢                                  ⁢        5            
It is again apparent from this relationship that thermally induced changes to the resonant frequency of a beam may be retarded (i.e., compensated for) or enhanced by changes in an elastic strain, (d(S)), applied to the beam.
Unfortunately, the thermal coefficient of Young's modulus for silicon is in the order of 30 ppm/K. This reality leads to considerable temperature drift in the frequency of an oscillating beam in the range of 18 ppm/C°. Given nominal requirements for temperature stabilities ranging from 0.1 to 50 ppm, and common operating temperature specifications ranging from −40° C. to +85° C., the putative MEMS designer faces a considerable challenge in the design of a temperature stable resonator.
Clearly, an efficient compensation mechanism is required for frequency stability of micromechanical resonators over an operating temperature range. Such a mechanism should not rely on the incorporation of materials incompatible with CMOS integrations.